To simplify Ah(x,y), put k' = k â h + 2 + 2x, and then let m = 2x + 1, ... E. Kay, H. Soul and D. Trutt, Some subnormal operators and hypergeometric kernel ...
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 66, Number
2. October
1977
SUBNORMALGENERALIZEDHAUSDORFF OPERATORS B. K. GHOSH, B. E. RHOADES AND D. TRUTT
Abstract. One type of generalized Hausdorff matrix is the lower triangular matrix with entries h„k = ("„-%)&!'~k»k,where Ae„ = vn — v„+l, vn = J'¿7"+ 0. The matrix Ha generated by v„ = (n + a + 1)_1 is shown to be a subnormal operator on I2 if a is a nonnegative integer.
1. Introduction and summary. One type of generalized Hausdorff matrix is the lower triangular matrix with entries
*.-(::
-ky-\.
where Ar, -»■-
Vu-
vn=(\"+adß(t)
for some ß(t) of bounded variation on [0, 1] and for some a > 0 (see [4]). Let Ha denote the generalized Hausdorff matrix Then hnk = (n + a + l)~x, so that Ha is a triangular matrix A = (ank) is called factorable [7, Theorem 2], any matrix which formally
generated by vn = fx0t"+a dt. factorable matrix. (A lower if a„¿ = bnck when k < n.) By commutes with Ha must be
triangular and, hence, by the proof of Theorem 198 in [3], must also be a generalized Hausdorff matrix. Thus, the operators on I2 which commute with Ha are all generalized Hausdorff matrices. (For each a, the commutant of Ha is a closed algebra of operators. For a ¥= ß, the only operators which commute with both Ha and Hß are the zero operator and the identity operator.) Our main result is that, for every nonnegative integer a, Ha is a subnormal operator on I1 and hence, by Yoshino's theorem [8] and the comments above, so is any generalized Hausdorff matrix which maps I2 into I2 and commutes with Ha. We explicitly describe a Borel probability measure ua on the closed unit disk (actually on the circles \z — n/(n + 1)| = \/(n + 1)) such that / - Ha is unitarily equivalent to the operator/(z) -^ zf(z) acting on H2(pa), the closure of the polynomials in the metric of L2( jua). For a = 0, this result was obtained in [5], [6]. For a a positive integer, the result is new. We are unable to determine whether Ha is subnormal for nonnegative noninteger values of a. Our approach relies on the methods and results of [1], [5], [6]. Received by the editors September 30, 1976.
AMS (MOS) subjectclassifications (1970).Primary33A15,33A30,33A65,40G05,47B20. © American Mathematical Society 1977
261
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262
B. K. GHOSH, B. E. RHOADES AND D. TRUTT
2. Let a > 0. We consider Ha as a linear transformation on l2 or, equivalently, on the Hardy space H2 of functions/(z) = 1,anz" analytic for \z\ < 1 with 2|a„|2 < oo. Lemma 1. For each a > 0, Ha is a bounded linear operator on I2 with
Il//„Il = 2. The point spectrum of H* is the disk [X: |1 - X\ < 1). Each X in this disk is a simple eigenvalue with corresponding eigenfunction —\.(We will retain the notation h = a +
1 throughout.) Lemma 3. For all real numbers a, b and nonnegative integers x,
" A
T(a+\)T(b+j) ß (m -J)xT(b)T(a -m+j+l)
_ ' (
where we interpret l/T(-w + 1) = 0 and T(-c 1) ... (— c + n — I) for all positive integers n.
T(b-a + m) V) m\T(b - a) '
+ n)/T(-c)
= (-c)(-c
+
Proof. The identity follows if one equates the coefficients of tm in the binomial expansions of both sides of the identity (I — t)P(l - t)~b = (I — ty(b-a)
Corollary.
»-¿-1,
y±o l
If h > k are nonnegative integers, then
ni
Tjh + k+j)
j ß (k +jy.T(h -j-k)
k_x T(h + k)
"K
}
k\T(h - k) ■
Proof. Let a = h - 1, b = h + k and n = h - k - \ in Lemma 3. The result follows. Lemma 4. For h a positive integer and \c\ < I, we have the identity
7,-1 (_i)A-*-'r(/j
+ k\
F(h, h;l;c) = (l- c)-" 2 —~2-"-" Jfc-0
(1 - 0
•
(/Cl )2r(/7 - /C)
Proof. Using formulas (4) on p. 57 and (22) on p. 64 of [2], and the
identity r(7i)r(l - h)/T(l - h + j) = (- iyT(A - j), we get
F(h, h; 1; c) = (1 - c)-"F(h, 1 - h; 1; c/ (c - 1))
Expand (c/(l - c)y = (1/(1 - c) - iy by the binomial theorem and set k = j - i. The result then follows from the Corollary. Theorem 1. For each positive integer h and all 0 < a, b < 2,
(ab)"-lF(h, h, 1; (1 - a)(l -b)) = f •* — CO'
»v/tere ßh(x,y)
j
aV dßh(x,y), — 00
is a joint probability measure defined by
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264
B. K. GHOSH, B. E. RHOADES AND D. TRUTT
|r(x + 1 + iy)f dydp(x) dßh(x,y)
=
2trT2(h)T(2x+ 2)\T(x - h + 2 + iy)\ where p(x) = \ if x = (n — l)/2 for some n = 0, 1, 2, 3, ... otherwise.
, and p(x) = 0
Proof. By Lemma 4,
(ab)"-xF(h,h,
\; (\ - a)(\ - b)) u ,*«'
(~l)h~k~]T(h
= (ab)1-1 Y
+ k)
--'—-
k=o
(h+k)
(a + b - ab)-(h+k).
(k\ )2T(h - k)
By Theorem 1 in [5],
, ^ u
m- 1, and (c)0 = 1. Putting m = h
— I, a = x + I + iy, ß = x + I — iy, c = 2x — h + 3, and noting T(h —k) = (-\)kT(h)/(-h + l)k, it follows that Bk(x,y)
=\T(x
+ 1 + 7»|4{r2(/7)r(2x
+ 2)|r(*
- h + 2 + iy)\2}~\
To simplify Ah(x,y), put k' = k — h + 2 + 2x, and then let m = 2x + 1, a = h — 1 — x + iy, ß = h — 1— x — iy, c = h —2x — lin Saalschiitz's Theorem to get Ah(x,y) -\X(x + 1 + iy)\2\T(h -\-x x {r2(/7)r(2x + 2)|r(-x
+ iy)\2 + 7»|2}~'.
It follows that Ah(x,y) = Bh(x,y), which establishes the theorem except for the verification that ßh is a probability measure. But this follows from the observation that
2 P
^-Bh(x,y)dy = F(h, A,1;0) = 1 and 5A(*.>0>0-
•'-co
Theorem subnormal.
^f
2. // a = 0, 1, 2, 3,. . .,
then the operator I — Ha on I2 is
Proof. The proof is essentially the same as the proof of Theorem 2 in [5]. Remark. The formula for ßh(x,y) in Theorem 1 defines a positive measure whenever h > 1. However, the functions
